Several topics between geometric measure theory, singular integrals and harmonic measure
The aim of this minicourse is to present several results that played a crucial role in the development of several fields of modern analysis.
One group of those was obtained by Peter Jones in eighties and, essentially, concerns the L2 boundedness of the Cauchy transform on Lipschitz graphs. This particular result has quite an interesting and rich story, one can say that it began some fifty years before its conception, and then branched into several areas of analysis providing an immense impact in (among other things) PDEs, harmonic measure, harmonic analysis, approximation theory and quite a few others.We are going to talk about one of its parts which concerns the so-called Jones beta numbers and his own proof of the aforementioned L2 boundedness with some extra applications and recent developments. Special attention will be given to proofs and elucidating various arising techniques.
The second group is focused on one of the extremely important counterexamples regarding the boundary values of harmonic functions and 3d version of the Privalov theorem which was given by Wolff around thirty years ago (still, in a sense, the best known result to this day). Here we will also discuss some applications to the harmonic measure properties and a variant of the Beurling circular projection theorem.
One group of those was obtained by Peter Jones in eighties and, essentially, concerns the L2 boundedness of the Cauchy transform on Lipschitz graphs. This particular result has quite an interesting and rich story, one can say that it began some fifty years before its conception, and then branched into several areas of analysis providing an immense impact in (among other things) PDEs, harmonic measure, harmonic analysis, approximation theory and quite a few others.We are going to talk about one of its parts which concerns the so-called Jones beta numbers and his own proof of the aforementioned L2 boundedness with some extra applications and recent developments. Special attention will be given to proofs and elucidating various arising techniques.
The second group is focused on one of the extremely important counterexamples regarding the boundary values of harmonic functions and 3d version of the Privalov theorem which was given by Wolff around thirty years ago (still, in a sense, the best known result to this day). Here we will also discuss some applications to the harmonic measure properties and a variant of the Beurling circular projection theorem.
日期
2025年06月11日 至 07月09日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周一,周三 | 13:30 - 15:05 | A3-4-101 | ZOOM 14 | 712 322 9571 | BIMSA |
修课要求
The listener should be acquainted with basics of real analysis, functional analysis and probability, some knowledge of harmonic functions and Hilbert transform is also recommended. Nevertheless we attempt to present most of the proofs in a self-contained way.
课程大纲
i. Introduction: brief account of the history, Hilbert transforms, Cauchy operators and approximation problems.
ii. Travelling salesman and rectifiable sets: the beta numbers.
iii. Rectifiable sets have finite beta value.
iv. How to cover a set with finite beta value by a rectifiable curve.
v. L2 boundedness of the Cauchy integral.
vi. Some further comments and appilcations.
vii. Wolff's example for Privalov's theorem in three dimensions.
viii. Circular projection theorems.
ii. Travelling salesman and rectifiable sets: the beta numbers.
iii. Rectifiable sets have finite beta value.
iv. How to cover a set with finite beta value by a rectifiable curve.
v. L2 boundedness of the Cauchy integral.
vi. Some further comments and appilcations.
vii. Wolff's example for Privalov's theorem in three dimensions.
viii. Circular projection theorems.
参考资料
1) J. Azzam and R. Schul. An Analyst’s traveling salesman theorem for sets of dimension larger than one. Math. Ann. 2018 Apr;370(3-4):1389-476.
2) C.J. Bishop and P.W. Jones. Harmonic measure, L2-estimates and the Schwarzian derivative. Journal D’analyse Mathematique. 1994 Dec;62(1):77-113.
3) J.B. Garnett, D.E. Marshall. Harmonic measure. Cambridge University Press, (2005), 571 pp.
4) Immo Hahlomaa. Menger curvature and Lipschitz parametrizations in metric spaces. Fund. Math., pages 143–169, 2005.
5) Peter W. Jones. Rectifiable sets and the traveling salesman problem. Invent. Math., 102(1):1–15, 1990.
6) P. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, in: “Proc. Conf. on Harmonic Analysis and Partial Differential
Equations”, El Escorial 1987 (J. Garc´ıa-Cuerva, ed.), Lecture Notes in Math. 1384, Springer-Verlag, 1989, pp. 24–68
7) B. Jaye, X. Tolsa, and M. Villa. A proof of Carleson’s ε2-conjecture. Ann. of Math. (2) 194 (2021), no. 1, 97–161, P. Mattila. Rectifiability: a survey, arXiv:2112.00540v1.6
8) H. Pajot, Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral, Lecture Notes in Math. 1799, Springer-Verlag, Berlin Heidelberg, 2002
9) J. Verdera, Birth and life of the L2 boundedness of the Cauchy Integral on Lipschitz graphs, arXiv:2109.06690.
10) T. H. Wolff, Counterexamples with harmonic gradients in R3, pp. 321–384 in Essays on Fourier analysis in honor of Elias M. Stein (Princeton, 1991), Princeton Math. Ser. 42, Princeton Univ. Press, Princeton, NJ, 1995
2) C.J. Bishop and P.W. Jones. Harmonic measure, L2-estimates and the Schwarzian derivative. Journal D’analyse Mathematique. 1994 Dec;62(1):77-113.
3) J.B. Garnett, D.E. Marshall. Harmonic measure. Cambridge University Press, (2005), 571 pp.
4) Immo Hahlomaa. Menger curvature and Lipschitz parametrizations in metric spaces. Fund. Math., pages 143–169, 2005.
5) Peter W. Jones. Rectifiable sets and the traveling salesman problem. Invent. Math., 102(1):1–15, 1990.
6) P. Jones, Square functions, Cauchy integrals, analytic capacity, and harmonic measure, in: “Proc. Conf. on Harmonic Analysis and Partial Differential
Equations”, El Escorial 1987 (J. Garc´ıa-Cuerva, ed.), Lecture Notes in Math. 1384, Springer-Verlag, 1989, pp. 24–68
7) B. Jaye, X. Tolsa, and M. Villa. A proof of Carleson’s ε2-conjecture. Ann. of Math. (2) 194 (2021), no. 1, 97–161, P. Mattila. Rectifiability: a survey, arXiv:2112.00540v1.6
8) H. Pajot, Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral, Lecture Notes in Math. 1799, Springer-Verlag, Berlin Heidelberg, 2002
9) J. Verdera, Birth and life of the L2 boundedness of the Cauchy Integral on Lipschitz graphs, arXiv:2109.06690.
10) T. H. Wolff, Counterexamples with harmonic gradients in R3, pp. 321–384 in Essays on Fourier analysis in honor of Elias M. Stein (Princeton, 1991), Princeton Math. Ser. 42, Princeton Univ. Press, Princeton, NJ, 1995
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笔记公开
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语言
英文
讲师介绍
Pavel Mozolyako is an associate professor at St. Petersburg State University. He leads PhD program in mathematics at the department of Mathematics and Computer Science. He got his PhD degree in 2009, at St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences. He was a postdoc at Norwegian University of Science and Technology, University of Bologna, and a visiting professor at Michigan State University. His research considers mostly boundary behaviour of harmonic functions and discrete models in potential theory.